MinT - Best Known (210−39, 210, s)-Nets in Base 3

Best Known (210−39, 210, s)-Nets in Base 3

(210−39, 210, 896)-Net over F₃ — Constructive and digital

Digital (171, 210, 896)-net over F₃, using

3² times duplication [i] based on digital (169, 208, 896)-net over F₃, using
- trace code for nets [i] based on digital (13, 52, 224)-net over F₈₁, using
  - net from sequence [i] based on digital (13, 223)-sequence over F₈₁, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 13 and N(F) ≥ 224, using
      - a function field by SÃ©mirat [i]

(210−39, 210, 3595)-Net over F₃ — Digital

Digital (171, 210, 3595)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3²¹⁰, 3595, F₃, 39) (dual of [3595, 3385, 40]-code), using
- discarding factors / shortening the dual code based on linear OA(3²¹⁰, 6579, F₃, 39) (dual of [6579, 6369, 40]-code), using
  - constructionÂ X applied to C([0,19]) âŠ‚ C([0,18]) [i] based on
    1. linear OA(3²⁰⁹, 6562, F₃, 39) (dual of [6562, 6353, 40]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562Â | 3¹⁶âˆ’1, defining interval IÂ =Â [0,19], and minimum distance dÂ â‰¥ |{−19,−18,…,19}|+1Â =Â 40 (BCH-bound) [i]
    2. linear OA(3¹⁹³, 6562, F₃, 37) (dual of [6562, 6369, 38]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562Â | 3¹⁶âˆ’1, defining interval IÂ =Â [0,18], and minimum distance dÂ â‰¥ |{−18,−17,…,18}|+1Â =Â 38 (BCH-bound) [i]
    3. linear OA(3¹, 17, F₃, 1) (dual of [17, 16, 2]-code), using
      - discarding factors / shortening the dual code based on linear OA(3¹, s, F₃, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
        parity-check code [i]

(210−39, 210, 702276)-Net in Base 3 — Upper bound on s

There is no (171, 210, 702277)-net in base 3, because

1 times m-reduction [i] would yield (171, 209, 702277)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 5228 211180 483822 644205 992035 939119 304670 434471 382172 838799 849822 069378 222041 752344 191819 518041 663019 > 3²⁰⁹ [i]